Risk measurement and Implied volatility under Minimal Entropy Martingale Measure for Levy process

This paper focuses on two main issues that are based on two important concepts: exponential Levy process and minimal entropy martingale measure. First, we intend to obtain   risk measurement such as value-at-risk (VaR) and conditional value-at-risk (CvaR) using Monte-Carlo methodunder minimal entropy martingale measure (MEMM) for exponential Levy process. This Martingale measure is used for the exponential type of the processes such as exponential Levy process. Also, it can be said MEMM is a kind of important sampling method where the probability measure with minimal relative entropy replaces the main probability. Then we are going to obtain VaR and CVaR by Monte-Carlo simulation. For this purpose, we have to calculate option price, implied volatility and returns under MEMM and then obtain risk measurement by proposed algorithm. Finally, this model is simulated for exponential variance gamma process. Next, we intend to develop two theorems for implied volatility under minimal entropy martingale measure by examining the conditions. These theorems consider the asymptotic implied volatility for the case that time to maturity tends to zero and infinity.